
// Make newform 2116.4.a.g in Magma, downloaded from the LMFDB on 30 May 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_2116_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_2116_4_a_g();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_2116_4_a_g();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [-4554, -38502, -83457, -6114, 87865, 12114, -23617, -2074, 2363, 122, -91, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [899155827903546, 3944408158791402, 1550634321988278, -3958905488112916, -1192704270961065, 1111822726899067, 214685342663884, -123544397554238, -12899275921940, 5331816250018, 196889362019, -62613704029], [-647911800243282432, -1836493640118024942, -293055683999121081, 2532788911144383202, -46764895849079646, -681851966177681614, 69012683278130201, 51841502899425998, -5769547369816246, -1129950561222082, 82637527027135, 6319786649326], [-480579521613083673, -1126838266426625154, 466700548994006841, 659458685754411338, 10385194496190138, -130033052718945356, -24815044068221393, 14220846023281678, 2073124411266004, -685891930550582, -33495022894813, 8595448175360], [-232103525519318310, -3590105580917170854, -2256850060434027814, 4292891580944366084, 637195812462780367, -1146927963985661233, 10652471331372860, 92009028658794746, -4196005379291152, -2371884766106694, 66187120234507, 18050413258583], [-78926374722490020, -1355799473950770552, -943426765039193100, 1753830976555138640, 474316531302392010, -496145123436143375, -63627001225358780, 49935853005133714, 3214658455774630, -1911939640064690, -48252679082140, 20963799939023], [161168048171778142, 763105854757815936, 318628669768748451, -859616150530813653, -160280525041902667, 234719087355232551, 18198617239779150, -22555516645346300, -814035680032991, 814038886355121, 12891554737513, -8567536578199], [-1120155687551114874, -4420926860951297730, -856821064669367870, 4793446342760724874, 664469725745016335, -1289339146933701728, -96205097952031376, 126372540743268130, 5271178826703478, -4730443845519672, -87680552899219, 51030168465448], [-326512852605500022, -1721414407081216056, -1123578659441800814, 2093818650919358133, 622397913524964311, -594197977946584393, -87956064821384782, 60996407769298787, 4576145504669582, -2393077927404176, -68197395581591, 26646411492390], [186115115326487325, 1653357021566353086, 1592461066757496669, -2214706698615565678, -855518106221543052, 677150247009777286, 131046514705758391, -75472386601095110, -7291749908062436, 3185006666962486, 109446653930981, -36862100949694], [-637550993180145363, -3245125897090096152, -1456590378011464647, 3849056982330371501, 879534496611442701, -1093127806594714631, -124114259188947098, 111432349776945076, 6551068167032449, -4307862995993705, -101806962244177, 47396489450567], [2509778722870502424, 11472137971747809444, 5443953897257929688, -12693903843577028108, -3564301453708661132, 3568225460368710825, 572144019580287132, -378375780655975062, -32389641684354990, 15489320058831490, 492455576018442, -176739163936469]];
    Rf_basisdens := [1, 114938065137162, 6953752940798301, 6953752940798301, 13907505881596602, 13907505881596602, 4635835293865534, 13907505881596602, 6953752940798301, 6953752940798301, 6953752940798301, 13907505881596602];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_2116_a();"
function MakeCharacter_2116_a()
    N := 2116;
    order := 1;
    char_gens := [1059, 5];
    v := [1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_2116_a_Hecke( : Kf := Kf)
    return MakeCharacter_2116_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 4;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, -1, 0, 0, 1, 0, 0, 0, -1, 0, 0, 1], [7, 0, 0, 1, 0, 0, -2, 0, 0, 0, 1, 0], [0, 6, 0, 0, 1, 3, 0, 1, 0, 0, 0, 2], [0, 20, 0, 0, -1, 2, 0, -1, 1, 0, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-8, 0, 3, 1, 0, 0, 16, 0, 0, 3, -1, 0], [29, 0, 3, -1, 0, 0, -16, 0, 0, -2, 4, 0], [0, -11, 0, 0, 4, -9, 0, 1, 21, 0, 0, 0], [54, 0, -2, 3, 0, 0, 10, 0, 0, -4, 10, 0], [0, -19, 0, 0, -2, 12, 0, 7, -14, 0, 0, 3], [31, 0, 4, 3, 0, 0, -37, 0, 0, -14, 0, 0], [0, 46, 0, 0, -4, -13, 0, 15, 14, 0, 0, 2], [63, 0, -5, -3, 0, 0, -57, 0, 0, 16, -2, 0], [0, 49, 0, 0, 0, 25, 0, 0, 23, 0, 0, -5], [0, 85, 0, 0, -13, -39, 0, -8, 17, 0, 0, -9], [114, 0, -5, -10, 0, 0, -31, 0, 0, 28, -6, 0], [7, 0, -5, -3, 0, 0, -40, 0, 0, -18, -5, 0], [0, 150, 0, 0, 18, -16, 0, 8, -14, 0, 0, 14], [0, -25, 0, 0, -18, -50, 0, 9, 30, 0, 0, -1], [0, -239, 0, 0, 5, 54, 0, -15, 8, 0, 0, -9], [0, -151, 0, 0, 1, -33, 0, -9, -8, 0, 0, 24], [65, 0, -15, 6, 0, 0, -2, 0, 0, 2, 12, 0], [0, -182, 0, 0, 15, -77, 0, 9, 31, 0, 0, 2], [0, 319, 0, 0, -12, 37, 0, 7, -40, 0, 0, 21], [0, 267, 0, 0, 2, 106, 0, -40, 29, 0, 0, -2], [0, 192, 0, 0, -8, -3, 0, 41, -31, 0, 0, 14], [73, 0, -17, -31, 0, 0, -10, 0, 0, 18, -46, 0], [773, 0, 1, -9, 0, 0, -75, 0, 0, 56, -14, 0], [0, -45, 0, 0, -36, 17, 0, -36, 78, 0, 0, -1], [1494, 0, 20, 25, 0, 0, -62, 0, 0, 28, 15, 0], [0, -487, 0, 0, -10, -109, 0, 30, -63, 0, 0, 11], [-25, 0, 38, 7, 0, 0, -71, 0, 0, -14, -16, 0], [0, 301, 0, 0, -10, -64, 0, -10, -151, 0, 0, -26], [38, 0, 46, 37, 0, 0, -27, 0, 0, 4, -17, 0], [427, 0, -13, 65, 0, 0, -112, 0, 0, -30, 12, 0], [1564, 0, -16, -6, 0, 0, -20, 0, 0, 59, 26, 0], [-1258, 0, -34, 13, 0, 0, -74, 0, 0, -64, 71, 0], [0, -771, 0, 0, -10, 54, 0, -11, 63, 0, 0, 33], [0, 50, 0, 0, -5, 89, 0, -51, -161, 0, 0, -10], [-79, 0, -15, 19, 0, 0, 252, 0, 0, -14, 51, 0], [832, 0, -35, 27, 0, 0, 124, 0, 0, -87, 51, 0], [0, 551, 0, 0, 53, -111, 0, -30, -11, 0, 0, 12], [326, 0, -40, -46, 0, 0, -149, 0, 0, 152, -36, 0], [1897, 0, -66, -5, 0, 0, -143, 0, 0, 34, -26, 0], [0, -947, 0, 0, -33, -199, 0, 108, -115, 0, 0, 33], [0, -655, 0, 0, 34, -189, 0, 67, 193, 0, 0, 26], [1742, 0, 125, 40, 0, 0, 2, 0, 0, -13, -14, 0], [422, 0, 50, -12, 0, 0, -90, 0, 0, 24, 38, 0], [0, -752, 0, 0, -8, 157, 0, -71, 71, 0, 0, 50], [0, 1100, 0, 0, -33, -76, 0, 55, 77, 0, 0, -36], [1808, 0, -28, 77, 0, 0, 478, 0, 0, -99, 29, 0], [0, 2333, 0, 0, 6, -140, 0, 37, -226, 0, 0, -10], [2545, 0, 23, -48, 0, 0, 614, 0, 0, -5, -78, 0], [-3489, 0, -36, 19, 0, 0, -329, 0, 0, 126, 36, 0], [4967, 0, 17, 8, 0, 0, 158, 0, 0, 51, -36, 0], [0, -1866, 0, 0, -33, 344, 0, 61, 38, 0, 0, 17], [0, 107, 0, 0, 56, 380, 0, 79, -68, 0, 0, 41], [0, 1233, 0, 0, 34, -154, 0, 108, 211, 0, 0, -98], [-1874, 0, 10, 118, 0, 0, 301, 0, 0, 36, 64, 0], [-574, 0, 85, 60, 0, 0, 307, 0, 0, 80, -132, 0], [0, -709, 0, 0, 10, -24, 0, 12, -32, 0, 0, 100], [-3301, 0, -110, -33, 0, 0, -38, 0, 0, 12, -71, 0], [-1874, 0, 56, 97, 0, 0, -388, 0, 0, -312, -9, 0], [0, 2457, 0, 0, 46, -225, 0, 22, 120, 0, 0, 171], [2324, 0, -34, -18, 0, 0, 257, 0, 0, 32, 98, 0], [1907, 0, -2, -119, 0, 0, 718, 0, 0, -65, -109, 0], [-898, 0, -40, -43, 0, 0, 510, 0, 0, 155, -37, 0], [0, -1657, 0, 0, -86, 254, 0, 167, 378, 0, 0, 36], [0, -1157, 0, 0, -115, 201, 0, -138, -227, 0, 0, 14], [0, -1745, 0, 0, -78, 182, 0, -81, 121, 0, 0, 27], [0, -96, 0, 0, 95, 574, 0, 31, 237, 0, 0, 126], [0, 74, 0, 0, 133, -419, 0, -29, 77, 0, 0, -150], [0, 137, 0, 0, 194, 422, 0, 301, -97, 0, 0, 113], [343, 0, 117, 212, 0, 0, 326, 0, 0, -84, 34, 0], [0, 2694, 0, 0, -32, 204, 0, -489, 49, 0, 0, -67], [-1370, 0, 133, -19, 0, 0, 448, 0, 0, 241, 54, 0], [0, -1593, 0, 0, -39, 137, 0, -88, -473, 0, 0, 173], [0, 1204, 0, 0, 50, -106, 0, -206, 194, 0, 0, -124], [0, 1377, 0, 0, -61, 115, 0, 50, 63, 0, 0, 242], [0, 1651, 0, 0, 87, -123, 0, 317, -424, 0, 0, 340], [311, 0, 89, 51, 0, 0, 538, 0, 0, 418, 6, 0], [3376, 0, -198, -65, 0, 0, -196, 0, 0, 160, 35, 0], [4340, 0, 132, 151, 0, 0, -254, 0, 0, -19, 119, 0], [0, -2694, 0, 0, -147, 46, 0, 50, -121, 0, 0, 78], [4705, 0, -207, -160, 0, 0, -870, 0, 0, -20, -6, 0], [4988, 0, -24, 68, 0, 0, 546, 0, 0, -24, -8, 0], [0, -1020, 0, 0, -26, 456, 0, -14, 190, 0, 0, 24], [0, 784, 0, 0, 157, 465, 0, 105, -7, 0, 0, 104], [-4248, 0, -41, 4, 0, 0, -1183, 0, 0, 152, 30, 0], [1457, 0, 157, -21, 0, 0, -1286, 0, 0, -92, 254, 0], [645, 0, 39, -256, 0, 0, 432, 0, 0, 36, -129, 0], [0, 1697, 0, 0, 13, -175, 0, -60, 277, 0, 0, -106], [-3054, 0, 98, 252, 0, 0, 1040, 0, 0, -211, 120, 0], [0, 1367, 0, 0, -132, -286, 0, 245, -559, 0, 0, -47], [0, -4008, 0, 0, 26, 624, 0, -322, -270, 0, 0, -212], [3610, 0, 0, 166, 0, 0, -512, 0, 0, -87, 18, 0], [-949, 0, -5, -263, 0, 0, 930, 0, 0, 384, 30, 0], [0, 567, 0, 0, 14, -399, 0, 24, -339, 0, 0, -133], [0, -1829, 0, 0, 366, -16, 0, -35, -610, 0, 0, -65], [0, 1889, 0, 0, -36, 507, 0, 464, -638, 0, 0, -35], [0, 284, 0, 0, 32, 265, 0, -440, -1324, 0, 0, -106], [1847, 0, -193, -368, 0, 0, 342, 0, 0, 96, -133, 0], [-6849, 0, 157, -380, 0, 0, 832, 0, 0, 128, -87, 0], [3492, 0, -81, -39, 0, 0, 232, 0, 0, 65, -368, 0], [-6840, 0, -113, 40, 0, 0, -1199, 0, 0, 528, 22, 0], [-5760, 0, 149, 76, 0, 0, 746, 0, 0, -170, 191, 0], [2348, 0, -137, 146, 0, 0, -607, 0, 0, 52, 252, 0], [0, -989, 0, 0, 4, 691, 0, -142, 405, 0, 0, -245], [0, -5619, 0, 0, -123, 47, 0, -462, 477, 0, 0, -415], [0, -402, 0, 0, 162, -648, 0, 140, -342, 0, 0, 352], [0, -256, 0, 0, -145, -1027, 0, 147, 1307, 0, 0, -64], [0, 1549, 0, 0, -178, -65, 0, -606, 54, 0, 0, -13], [0, 2271, 0, 0, 78, 669, 0, 109, 598, 0, 0, 103], [-4057, 0, -26, -245, 0, 0, 1415, 0, 0, -70, -308, 0], [5038, 0, -11, 145, 0, 0, 1128, 0, 0, 417, 69, 0], [0, -1627, 0, 0, 113, -401, 0, 306, -881, 0, 0, 503], [0, 3329, 0, 0, 120, 1603, 0, 176, -9, 0, 0, 325], [-2734, 0, -48, 60, 0, 0, 920, 0, 0, -561, 89, 0], [0, -5558, 0, 0, -140, -552, 0, 313, -720, 0, 0, -125], [-6203, 0, 11, -30, 0, 0, -1372, 0, 0, 648, -379, 0], [-2479, 0, -143, 33, 0, 0, -884, 0, 0, -420, 684, 0], [0, -1697, 0, 0, 86, 875, 0, 289, -753, 0, 0, 448], [0, -3716, 0, 0, 162, -521, 0, -105, 796, 0, 0, 42], [-2975, 0, -15, -291, 0, 0, 1326, 0, 0, 186, -458, 0], [0, 1795, 0, 0, -18, -834, 0, 535, -1070, 0, 0, 466], [0, -2785, 0, 0, -90, -456, 0, 140, 503, 0, 0, -476], [1732, 0, 86, 375, 0, 0, 1975, 0, 0, 448, 155, 0], [0, -3927, 0, 0, -197, 905, 0, 34, -253, 0, 0, 204], [0, -2533, 0, 0, -531, -503, 0, 598, -139, 0, 0, 230], [0, -5240, 0, 0, -36, -542, 0, 431, 760, 0, 0, 159], [4490, 0, -51, 216, 0, 0, 578, 0, 0, 397, 238, 0], [0, -12519, 0, 0, -140, 686, 0, -117, -205, 0, 0, -141], [0, 2808, 0, 0, 546, 88, 0, -366, 818, 0, 0, -174], [0, 4328, 0, 0, -142, 755, 0, 138, -214, 0, 0, -476], [0, 1661, 0, 0, -288, -49, 0, -1043, 169, 0, 0, -136], [12281, 0, 1, -176, 0, 0, 2038, 0, 0, -170, -365, 0], [-5942, 0, -296, 106, 0, 0, 321, 0, 0, 560, 196, 0], [2781, 0, -316, -299, 0, 0, 1454, 0, 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1604, 0, 0, 1000], [0, -38734, 0, 0, 1585, -11355, 0, 1129, 7972, 0, 0, -924], [-31043, 0, 1191, 1712, 0, 0, -8190, 0, 0, 6189, -3124, 0], [100946, 0, 1698, 728, 0, 0, 4307, 0, 0, 220, 446, 0], [13998, 0, -2228, 107, 0, 0, -535, 0, 0, -6320, -1867, 0], [0, 13283, 0, 0, 1754, 8750, 0, 6935, -6298, 0, 0, 1242], [0, 7523, 0, 0, -1742, -8404, 0, 3330, -7390, 0, 0, -1190], [0, -14252, 0, 0, -610, 6600, 0, 6070, -3034, 0, 0, -1096], [0, 13048, 0, 0, 1014, 2837, 0, 1981, -387, 0, 0, -610], [0, -7678, 0, 0, -1173, 132, 0, 1473, 7869, 0, 0, 170], [51977, 0, -3864, -1643, 0, 0, -4706, 0, 0, 1691, 2081, 0], [-4304, 0, -95, -963, 0, 0, -16212, 0, 0, 243, 3437, 0], [-27331, 0, 2448, 814, 0, 0, -10216, 0, 0, 5474, -1748, 0], [0, 35606, 0, 0, -1023, -10493, 0, -261, 5841, 0, 0, 1094], [0, 16373, 0, 0, -125, 7382, 0, -1525, -8098, 0, 0, 1637], [90273, 0, -1092, 895, 0, 0, -6846, 0, 0, 196, 561, 0], [-13096, 0, -2672, -452, 0, 0, -5102, 0, 0, -3704, 224, 0], [0, 11668, 0, 0, 2294, 4097, 0, -6059, 2988, 0, 0, -1166], [40038, 0, -1196, 1324, 0, 0, 2572, 0, 0, -3211, 75, 0], [0, 25891, 0, 0, -1159, 10791, 0, -1252, 7555, 0, 0, 141], [-52612, 0, 26, 1734, 0, 0, -11663, 0, 0, -472, 3930, 0], [-64997, 0, -2441, -40, 0, 0, -42, 0, 0, 5029, 386, 0], [43742, 0, 1320, 430, 0, 0, -3768, 0, 0, 1073, 1332, 0], [0, 19738, 0, 0, 3937, 8244, 0, -3617, 4383, 0, 0, -98], [0, 100817, 0, 0, 1640, -4750, 0, -887, 1296, 0, 0, 1733], [0, 33120, 0, 0, 601, -20, 0, -251, 2028, 0, 0, 585], [134101, 0, -811, 1950, 0, 0, -10090, 0, 0, -1213, 688, 0], [106272, 0, -959, -728, 0, 0, 7641, 0, 0, -3536, -750, 0], [73640, 0, -594, -658, 0, 0, 2264, 0, 0, 3740, -142, 0], [0, -19528, 0, 0, 2279, -7486, 0, -1129, 10044, 0, 0, -995], [0, 19783, 0, 0, 3140, -2028, 0, -4071, -3044, 0, 0, -1198], [0, 14477, 0, 0, -1692, -10551, 0, 1664, -5981, 0, 0, -1841], [0, 2636, 0, 0, -1390, 7212, 0, 4141, -10862, 0, 0, 1763], [23243, 0, 984, 2539, 0, 0, 5557, 0, 0, 3274, 3724, 0], [107717, 0, -1109, -284, 0, 0, 4282, 0, 0, -248, -617, 0], [-11933, 0, 949, 2773, 0, 0, -3708, 0, 0, 1405, 1716, 0], [82331, 0, 4103, 2102, 0, 0, -6230, 0, 0, 4405, -2172, 0], [0, -38488, 0, 0, -453, -708, 0, 4423, 12177, 0, 0, -478], [-44516, 0, 787, 12, 0, 0, -7234, 0, 0, 1237, -4842, 0], [-1582, 0, 2230, -2348, 0, 0, 6435, 0, 0, 7544, -2530, 0], [117051, 0, 685, -1841, 0, 0, -3620, 0, 0, -5410, -894, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2116_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2116_4_a_g();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_2116_4_a_g(:prec:=12)
    chi := MakeCharacter_2116_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_2116_4_a_g();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_2116_4_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2116_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![-440, 436, 2156, -104, -106, 1, 1]>,<5,R![70729402500, 0, -11172001167, 0, 626181309, 0, -15224262, 0, 168046, 0, -763, 0, 1]>],Snew);
    return Vf;
end function;